282 MOTION OF THE GAS SPHERE 



(8.11) differing by less than 2 per cent. It is therefore possible to con- 

 clude that this bubble period formula, and bubble period measurements, 

 can be relied upon to within two or three per cent if the charge is not 

 too close to boundary surfaces. The values of constant K for the first 

 bubble oscillations of several different explosives, obtained largely from 

 small charge data, are listed in Table 8.1, the units being chosen to give 

 the period in seconds for charge weight in pounds and depth in feet. 



C. Period of later oscillations and energy loss. The Willis formula 

 for period, Eq. (8.11), is based on the assumption that negligible energy 

 losses occur in the course of the bubble pulsation except at or very near 

 successive minima. Hence an expression of the same form should be 



1 Values corrected for surface effect by Friedman theory (Section 8.10). 



Table 8.1. Values of the constant K of the bubble period-depth relation 



T = KW'i^id -{- 33)5/6 for various explosives 



(T in seconds, W in pounds, d in feet). 



applicable to second and later pulsations if a new smaller value of Y is 

 used. The reason is simply that some of the energy in each cj^cle is 

 lost at each minimum in turbulence and radiation, and the next oscil- 

 lation occurs with lower energy. It is convenient to express this change 

 by rewriting the period relation in the form 



(8.12) 



n = 0.561 



irnQWy 



Zn 



5/6 



where W is the charge weight in pounds, rn is the fraction of the det- 

 onation energy Q which remains for the nth oscillation of period T n, 

 and 2 n is the equivalent depth in feet of water for the hydrostatic pres- 

 sure at the depth of the bubble at the beginning of the nth oscillation. 

 The value of 2;n will change significantly when the charge is fired rela- 

 tively shallow, and the distance of migration during a cycle is an ap- 

 preciable fraction of the depth. From Eq. (8.12), the ratio of the second 

 to first bubble periods is given by 



\-m) 



5/6 



